Greetings. So we're pushing forward with our trying to understand the theory of the firm, thinking about cost curves. We understand that cost curves come from a very specific outcome. When we have a solution for our fundamental problem here, is that the firm wants to maximize profits and profits are the residual of revenues and excess of cost. These are economic profits, so we've got an opportunity cost already built into that. So we did the total revenue side real easy, we understood how to do that from the days when we had a demand curve, at any price that the firm might charge and sell this output, then total revenue would just be equal to that price times that output. and if you took P_0 times Q_0, you're going to get the area of this rectangle, base times height. So we've got great graphical representations of that revenue side. But the bugaboo here is this, how do we find total cost? We understand that we find total cost by thinking about production costs come from production. Firms do not have magic elves that make the output overnight and they sell it the next day. People have to get up in the morning and make the donuts. People have to come to work and build cars. They won't do it for free, you've got to pay them. So we generated something last lecture that was a production function. That production function said look, the production function is a function of labor and capital. Those were our two inputs right now. The semicolon reminded us that this capitalists parametric to the problem. So in the short run, which is where we are, capitalist fixed, you can't change your brick and mortar today, but you can certainly have workers stay on and do over time. So we can vary labor anytime, but that's going to cost us money. Well, we could lay some off and that would save a little money, but what we're doing is trying to optimize here on making this profit maximization decision. So what we have to do is we need to think about costs, and costs are going to come to us in two sources. So we're going to write this out. Say the definition of total cost is the sum of fixed cost and variable cost. Definition of total cost is the sum of fixed cost and variable cost and fixed cost which by the way from now on, this is the last time you're going to see that notation. We're going to call this fixed costs and we're going to call this variable costs. So it's FC plus VC. What we have to do is we have to figure out how to graph this. Now, obviously from our previous example where we had the previous page, output was a function of labor and capitals parametric, its parametric. So this capital which is fixed is going to be captured by this fixed cost term. This labor, which of course is variable in the short run, is going to be captured by this variable cost term. But what we have do is figure out how to draw graph of this. We need to figure out a draft of this. So the way we're going to build this graph, is we're going to build a curve for the fixed cost, and then we're going to build the curve for the variable costs, and then we're going to do simple addition, because that's all it is; fixed cost plus variable cost will give us a total cost for the firm. So let's start drawing graphs. The fixed cost is pretty simple. So I'm going to put down an axis. On this axis, I'm going to measure dollars, and on this axis I'm going to measure output. Notice, the last couple axis's we've been doing, when production function we had on the horizontal axis inputs and on the vertical axis output. We switch that back down around now, because it's important for us to get output down here, so we get thinking about how to melt that in with our revenue side later on as we step forward to this. So fixed cost is pretty simple. There's a certain amount associated with whatever the cost of capital is, and that's our fixed cost curve, it's a horizontal line, simplest of all. Because by definition, it doesn't change. You can produce zero jars of mayonnaise, one jar of mayonnaise, 10 jars of mayonnaise, or a 100,000 jars of mayonnaise, and that Kraft plant and has the same fixed cost each day. That one was simple. Now we're going to get to a little bit harder step, we're going to get to something called variable cost. We're going to draw another axis system and we'll put again, consistent with what we did before, we're going to put up with there and we'll put output here. Now, I just want to remind you why we're doing this, because it's the beginning here. Notice, I said we've already got a revenue side and that's nice, but the revenue pictures got quantity on the horizontal axis and basically dollars and cents on the vertical axis. So we've got to start converting our life to that tune or thinking about costs. So the variable, when we think about variable, we're thinking about exactly how much do we have to pay for increasing our production? In other words, if we increase production, the only way we can increase production. The fixed cost was easy, it's there no matter what. But for variable costs, as we increase production, we have to hire more workers. We cannot just snap our fingers and wish more jars of mayonnaise. We've to hire more workers to come in, we have to get more eggs, more glass jars; all of these things pile up. How do we know about the relationship of that? Well, what we're going to do is we're going to bring back our old friend, the little thought bubble, above the cartoon characters head, there's a bubble that pops out of his head and says, "You know what. I seem to recall that earlier we looked at something that looked like this. As we hire inputs, we get outputs according to what's known as a standard production function, as short range of increasing returns, and then the law of diminishing marginal product sets in. So this is the production function with a fixed capital. Notice the generalized shape of that. It has this region where it is increasing and then it gets to an inflection point, and it gets to this region where it's flattening out. What that means is, as it's flattening out we're hiring lots and lots of extra workers to get diminishingly small growth in output, because of the law of diminishing marginal cost. So what we have to do is we have to figure out, how does this translate? What would be great is, if I was really good at this, I would have animation here and I would reach into this curve and grab the axis and flip them. When you were taking math courses and someone said, "I want you to take the inverse," They said, "Well, look. Here's a function. Y is equal to f of x." Then if the instructor said, "I want you to take the inverse of that," I say, "What's that mean?" You say, "Well, x is equal to the inverse f to the minus one of y." The first one, y is a function of x and it says, tell me what the amount x is and I'll tell you what the y is. If you invert it, it says, tell me how much you want for y and I can tell you how many x you need to have. If I could invert this production function, I would know that for any possible output I want, how many jars of mayonnaise do you want? For any possible jars of mayonnaise they want, I could by inverting it, I could know exactly how many different workers I need to hire. So the inverse of the production function, by inverting that production function, I can get labor back out. If I flipped it over, I'd now have a curve that showed me something like this. Bear with me now, we're almost out is, not the most fun. Now I got a function that's got labor on this axis, output on this axis, and it looks like this. It looks like that. This was the area of increasing returns, which meant that that cost curve, and if I could somehow translate, this man hours, these are physically against labor. But if I know the wage rate for that, I can scale this up by that scalar, whatever that is, and I've now got dollars and cents, and this becomes my variable cost. By scaling this up, this is actually physical man hours into that production function. It's all this, I just invert it. This is just the inverse of the production function. But if I scaled up by the cost, I now know what the variable cost is for every extra jar of mayonnaise I want to produce. This tells me how many workers, which I can translate by their wage rate, I'm actually going to have to pay for. So that means that's my variable cost. So let's recall where we were. A long time ago on this lecture, we wanted to find total cost. I said we're going to do it by finding fixed cost, which you did, that was easy, and we're going to add that to variable cost. So let's do that with one final picture. We'll do it by once again putting an axis system in place. On this axis I'm majoring dollars and cents, and on this axis I'm measuring output, and I have some amount of fixed cost, we drew this fixed cost curve, I'm just replicating it here, kind of in the background as a dotted function. Then we found out the variable cost function looks something like that. So its general shape looks like this. If I were to add those two, I would end up having a total cost curve which has a fixed cost component and it would just be basically the vertically displaced amount of variable cost, each one of these gaps between here is by definition or by construction, essentially the size of fixed cost. So this is our total cost curve. It has that general shape because of the law of diminishing marginal product, and that shape is going to help us to figure out, as we step forward, what marginal cost curves look like and that's going drive home our profit maximizing rule. Thanks. Thank you, Mike.